Geometric Objects as Platonic Forms

It is in fact true that no one has ever seen a true circle with their eyes.  When we explore the meaning of this seemingly preposterous proposition, we shall discover the right way to view mathematical objects, especially geometrical ones.  In the terminology of Platonic philosophy, there is a difference between the geometric objects perceived with the organs of physical sense-perception vs. the pure thought-picture which is apprehended by the mind alone.  This distinction is an exercise in discrimination between the real and the unreal, which is a main goal of Platonic education.

Plato, holding a manuscript of the Timaeus

In Plato's Timaeus, we find that sorting out the subtleties of this distinction is the necessary prerequisite for understanding the rest of Timaeus' exegesis describing the creation of the world.  In other words, unless we comprehend the difference between the material vs. the form/idea, then we are incapable of understanding anything else.

We must, then, in my judgment, first make this distinction: what is that which is always real and has no becoming, and what is that which is always becoming and never real?  That which is apprehensible by thought with a rational account is the thing that is always unchangeably real; whereas that which is the object of belief together with unreasoning sensation is the thing that becomes and passes away, but never has real being.
-- Plato, Timaeus (27D-28A)

The geometrical objects which we see with our eyes are not the real ones, since there is always some imperfection due to their being drawn with pencil or similar apparatus.  Euclid defines a line as "breadthless length".  However, upon closer scrutiny, a drawn line will be seen to have some measure of thickness, albeit on the scale of millimeters, but nonetheless it is not what Euclid meant by a line

A circle, but this not the Platonic form of the circle.

Therefore, a circle drawn on paper and seen with the eyes is not the real circle.  It is in the category of "the thing that becomes and passes away, but never has real being."  Eventually the paper on which the circle is drawn will decay and the drawn circle with it.  As an experiment, draw a nice circle on paper with a compass, then burn the paper.  You will then understand that, although the drawn circle has passed away, the idea of the circle remains.  This will help you perceive that the real circle is "that which is apprehensible by thought" and is "the thing which is always unchangeably real".

A similar thought is presented by the Neoplatonist Porphyry in his Life of Pythagoras:

As the geometricians cannot express incorporeal forms in words, and have recourse to drawings of figures, saying "This is a triangle", and yet do not mean that the actually seen lines are the triangle, but only what they represent, the knowledge in the mind, so the Pythagoreans used the same objective method in respect to first reasons and forms

Why then do mathematicians employ drawings in their work?  Because they are a powerful tool to lead the mind into the higher realm of the forms, like doorways that allow us to pass from the material world into the ideal world.  Practice with theoretical geometry is really an exercise in training the mind to gain freedom of mobility in this realm.  To this wrote Nicomachus of Gerasa in his Introduction to Arithmetic:

For it is clear that these studies are like ladders and bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those material, physical things, our foster-brethren known to us from childhood, to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls, and above all to the reason which is in our souls.

With patience and persistence, anyone can begin a study of mathematics as a spiritual discipline and find themselves in the world of ideas.  You will find that a new power of freedom opens inside the soul, a freedom which leads to real knowledge of the self, and to a true happiness that lasts.  We will continue exploring the methods of how this is achieved on Via Mathesis.